Method for joint modeling of mean and dispersion

ABSTRACT

The present invention describes a method and system for joint modeling of a mean and dispersion of data. A computing system derives a loss function taking into account distributional requirements over the data. The computing system represents separate regression functions for the mean and the dispersion as stagewise expansion forms. At this time, the stagewise expansion forms include undetermined scalar coefficients and undetermined basis functions. Then, the computing system chooses the basis functions that maximally correlate with a corresponding steepest-descent gradient direction of the loss function. The computing system obtains the scalar coefficients based on a single step of Newton iteration. The computing system completes the regression functions based on the chosen basis functions and obtained scalar coefficients.

BACKGROUND

The present invention generally relates to modeling data. Moreparticularly, the present invention relates to joint modeling of a meanand a dispersion of the data.

The joint modeling of a mean and dispersion refers to estimating themean and the dispersion concurrently. The estimated mean and dispersionmay or may not depend on each other. The (sample) mean is a measure ofthe central value of the data, and generally refers to an average valueamong sample data drawn from a population, e.g., a set of data aboutwhich statistical inferences are to be drawn. The (sample) dispersion isa measure of the spread of the data about its central value, and isgenerally measured by at least one or more of: (1) Range, (2) Meanabsolute deviation, (3) Standard deviation, (4) Variance and (5)Covariance, etc. The range refers to the difference between a largestvalue and a smallest value among the sample data. If the sample data iscomposed of x₁, x₂, . . . , x_(n), then the mean, m, is (x₁+x₂+ . . .+x_(n))/n. The mean absolute deviance is (|x₁−m|+|x₂−m|+ . . .+|x_(n)−m|)/n. The standard deviation is a square root of the variance,and the variance is {(x₁−m)²+(x₂−m)²+ . . . +(x_(n)−m)²}/n. When twovariables, x and y, are independent of each other, the covariance ofthem is 0. Generally, the variance of (x+y), Var(x+y), isVar(x)+Var(y)+2*Cov(xy), where Cov(xy) refers to the covariance of(x*y). Cov(xy) is {x₁y₁+x₂y₂+ . . . +x_(n)y_(n)}/n−mean(x)*mean(y),where x is {x1,x2, . . . , x_(n)}, y is {y₁,y₂, . . . , y_(n)}, n is thenumber of elements in x or y, mean(x) is the mean of x and mean(y) isthe mean of y. The covariate refers to a possibly predictive variablethat is related to an outcome of a modeling.

In regression applications (e.g., linear regression) involving aconditional response distribution from the exponential dispersionfamily, a primary interest is typically in a regression model for themean parameter. The corresponding dispersion parameter is either a fixedconstant (e.g., a unity for the Poisson and Bernoulli distributions), oris an unspecified constant that is estimated from residual deviancevalues of the mean regression model. The linear model refers to anyapproach to modeling relationship between one variable (or possibly morevariables) termed the response and denoted by y and one or morevariables termed the covariates denoted by x, such that the modeldepends linearly on an unknown parameters to be estimated from sampledata. The conditional response distribution refers to distributionalchanges of y across the sample data for fixed values of the covariatesx. A sample mean and variance (i.e., a measure of the central value andthe spread about the central value) are statistics computed from sampledata. The sample mean is an estimate of the mean parameter (μ) of thepopulation from which the sample data are drawn. The residual deviancevalue refers to a measurement of deviance contributed from each sampledata. Regression models are used to predict one variable, termed theresponse, from one or more other variables termed the covariates.Regression models provide a user with predictions about past, present orfuture events to be made with sample data about past or present events.Mean regression refers to a tendency that statistical outliers regresstoward the mean when sample data is tested over and over again.Regressing toward the mean refers to a relation between a value of avariable x and a value of y from which the most probable value of y canbe predicted for any value of x. Regression refers to a process fordetermining a line or curve that best represent a general trend ofsample data.

An assumption of constant dispersion implies that the mean and varianceof the conditional response distribution have some fixed and unchangingrelationship as a function of the covariates, even though in many casesthese quantities can plausibly vary quite independently. Althoughmodeling of a dispersion parameter (i.e., the sample dispersion is anestimate of a dispersion parameter (φ) of a population from which thesample data are drawn) may not be a primary goal of the regressionapplication, one consequence of any variability in the dispersionparameter is that non-constant case weights (i.e., weights for anon-constant dispersion parameter) are required in deviance lossfunctions used for the mean regression. Therefore, an accurateestimation of dispersion variability will result in tighter confidencebounds for parameter estimates in mean regression, as the dataassociated with the larger dispersion values will be appropriatelydown-weighted in terms of their contribution to the deviance lossfunction. However, the regression models for the mean and dispersionparameters cannot be obtained independently, or even sequentially, sincetheir estimation is intrinsically coupled in likelihood-basedformulations used for regression modeling. The deviance loss functionrefers to a function that that not only measures how close dispersionparameters are to their expected values, but also measures how welldispersion parameters correspond with dispersions of residuals. Thelikelihood-based formulations are functions of parameters of astatistical model that plays an important role in statistical inference.Statistical inference or statistical induction comprises uses ofstatistics and random sampling to make inferences concerning someunknown aspect of a population.

Traditionally, GLM (General Linear Model), which is widely used for meanregression modeling, has been used for conditional responsedistributions from the exponential dispersion family. The GLM is astatistical linear model for a suitable transformation of the mean, termthe link transformation. The GLM may be represented as g(Y)=XB+U, whereY is a vector with series of response measurements, g(•) is the linkfunction that is chosen appropriately for the assumed responsedistribution, X is a design matrix, B is a vector including parametersto be estimated, and U is a vector including errors and noises. Thedesign matrix refers to a matrix of explanatory variables (one orzeroes, or reals), that represents a specific statistical orexperimental model). However, a traditional methodology such as the GLMcannot perform joint modeling of the mean and the dispersion withoutinventing additional art, particularly in the case when the covariatesin the data are complex, and must be simplified and grouped in apreprocessing step that considerably detracts from the quality ofresulting model.

Therefore, it is highly desirable to provide a system and method toperform joint modeling of a mean and dispersion suitable for a widevariety of data sets without requiring any preprocessing and grouping ofthe sample data.

SUMMARY OF THE INVENTION

The present invention describes a system and method for perform jointmodeling of a mean and dispersion by extending MART (Multiple AdditiveTrees) methodology or extending the GLM methodology.

In one embodiment, there is provided a computer-implemented method forperforming joint modeling of a mean and dispersion of data, the methodcomprising:

deriving a suitable loss function characterized over the data takinginto account distributional details of the data;

representing regression functions for the mean and the dispersion asstagewise expansion forms, the stagewise expansion forms includingundetermined scalar coefficients and undetermined basis functions;

choosing the basis functions that maximally correlate with acorresponding steepest-descent gradient direction of the loss function;

obtaining the scalar coefficients based on a single step of Newtoniteration; and

completing the regression functions based on the chosen basis functionsand obtained scalar coefficients.

In one embodiment, there is provided a computer-implemented system Forperforming joint modeling of a mean and dispersion of data, the systemcomprising:

a memory device; and

a processor unit in communication with the memory device, the processorunit performs steps of:

deriving a suitable loss function characterized over the data takinginto account distributional details of the data;

representing regression functions for the mean and the dispersion asstagewise expansion forms, the stagewise expansion forms includingundetermined scalar coefficients and undetermined basis functions;

choosing the basis functions that maximally correlate with acorresponding steepest-descent gradient direction of the loss function;

obtaining the scalar coefficients based on a single step of Newtoniteration; and

completing the regression functions based on the chosen basis functionsand obtained scalar coefficients.

In a further embodiment, the processor unit further performs a step of:modifying parameters used for the obtaining the scalar coefficients toensure that the single step of the Newton iteration is a steep-descentgradient direction for the obtained scalar coefficients, to introducenumerical stability the computations.

In a further embodiment, the processor unit further performs a step of:modifying parameters used for the obtaining the scalar coefficients toachieve a decrease in the loss function for the obtained scalarcoefficients.

In a further embodiment, the chosen basis functions are regression treescomputed using a least-square splitting criterion.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings are included to provide a furtherunderstanding of the present invention, and are incorporated in andconstitute a part of this specification. The drawings illustrateembodiments of the invention and, together with the description, serveto explain the principles of the invention. In the drawings,

FIG. 1 illustrates a flow chart describing method steps for performing ajoint modeling of a mean and dispersion by extending GLM according toone embodiment of the present invention.

FIG. 2 illustrates a flow chart describing method steps for performing ajoint modeling of a mean and dispersion by extending MART methodologyaccording to one embodiment of the present invention.

FIG. 3 illustrates an exemplary hardware configuration used foroperating method steps described in FIG. 1 and/or FIG. 2 according toone embodiment of the present invention.

DETAILED DESCRIPTION

MART™ is an implementation of a gradient tree boosting method forpredictive data mining (regression and classification) described in“Greedy Function Approximation: A Gradient Boosting Machine”, J. H.Friedman, Annals of Statistics, Vol. 29 (5), 2001, pp. 1189-1232; whollyincorporated by reference as if set forth herein, hereinafter“Friedman”, describe the MART in detail. Gradient tree boosting methodrefers to computing a sequence of simple trees, where each successivetree is built for prediction residuals of a preceding tree.

According to one embodiment of the present invention, a computing system(e.g., a computing system 1600 depicted in FIG. 3) receives as inputprogrammed instructions for performing a joint modeling of a mean anddispersion by extending the MART methodology or extending GLMmethodology. Especially, the computing system 1600 performs jointregression of the mean and dispersion for a conditional responsedistribution (e.g., Inverse Gaussian distribution) from an exponentialdispersion family (i.e., Gamma, Normal and Inverse Gaussiandistributions). The joint regression of the mean and dispersion refersto estimating regression functions of the mean and dispersionconcurrently. A regression function describes a relationship between adependent variable Y and explanatory variable(s) X. The computing system1600 may estimate a regression function g( ) in a model, e.g., a linearmodel g(Y_(i))=X_(i)B+e_(i), where the e_(i) are residuals or errors.The Gamma, Normal and/or Inverse Gaussian distributions are continuousdistributions from the exponential distribution family for which thelikelihood takes a specific simple form, e.g., a likelihood function, l(), where:

${{l( {{y;\mu},\phi} )} = {\frac{1}{2}\lbrack {{\phi^{- 1}{d(\mu)}} - {\hat{s}( \phi^{- 1} )} - {\hat{t}(y)}} \rbrack}},$where φ is a dispersion parameter, d( ) is a deviance function thatreturns a deviance from a distribution with a mean parameter μ, and ŝ( )and {circumflex over (t)}( ) are certain functions such as log( ). Thecomputing system 1600 performs the joint regression by extending theMART methodology or the GLM to incorporate dispersion modeling (i.e.,estimating dispersion φ).

The present invention thus provides: (1) An easy incorporation ofrelevant nonlinear and low-order covariate interaction effects inregression functions by representing the regression functions aspiecewise-constant, additive and non-linear function (nonlinear effectsimplies that a covariate enters into a regression function not only as alinear term X_(i): but also as nonlinear terms such as X_(i) ² etc.;interaction effects are the nonlinear terms such as X_(i)X_(j), etc.)(2) Robust and computationally-efficient modeling procedures suitablefor large, high-dimensional data sets (high-dimensional refers to thenumber of covariates that may be in data, many of which may be collinearwith existing variables or irrelevant variables for a modeling of a meanor dispersion of the data) to the by using methods that intrinsicallyexclude collinear and irrelevant covariates while performing the jointmodeling (collinear covariates are equivalent to highly correlatedvariables that add no new useful information; irrelevant covariates haveno correlation with the response and are hence of no use for regressionmodeling); and (3) An ability to use high-cardinality categoricalcovariates, e.g., in insurance data sets there are covariates such aszip code information and car model information, etc., which can takeshundreds of values, and therefore cannot directly incorporated in theexisting regression methods without any preprocessing or grouping oflevels for this covariate, in order to have computational tractabilityin the modeling procedures. For example, the insurance data may includehigh-cardinality categorical features such as zip code, occupation code,car model type, etc. The computing system 1600 estimates regressionfunctions for a mean and dispersion of the insurance data withoutpreprocessing and grouping of the insurance data.

[1. Properties of Exponential Dispersion Family Distributions (EDs)]

A random variable Y˜ED*(θ, φ) from an univariate exponential dispersionfamily with a canonical parameter θ (i.e., a parameter which is afunction of the mean and a function of the variance; a canonicalparameter for Possion distribution is log_(e)(v) where v is a varianceof the distribution) and a dispersion parameter φ>0 has a probabilitydensity functionƒ(y;θ,φ)=exp{yθ−b(θ)−a(y)}/φ−c(y,φ)},  (1)where y∈Range (Y), b(θ) is called a cumulant function, and a(y) and c(y,θ) are certain functions such as formulas (15) that will be describedherein. A cumulant generating function (i.e., a function generatingcumulants of a random variable; a logarithm of a moment generatingfunction; the moment generating function of a random variable X isM_(x)(t):=E(e^(rX)) where r is a variable) of the formula (1) is givenbyK _(Y)(t)≡log E(e ^(tY))=b(θ+φt)−b(θ)/φ,  (2)where E( ) is a function calculating an expected value and t is aparameter, and from identities K′_(Y)(0)=E(y); K″_(Y)(0)=Var(Y), whereVar( ) is a variance function calculating a variance, we haveE(Y)=b′(θ), Var(Y)=φb″(θ).  (3)

Thus, if μ denotes the mean parameter, then from the formula (3)μ=b′(θ), where b′(•) is an invertible mean-value mapping (i.e., amapping between the canonical parameter and the mean parameter is aninvertible change of the parameters), and its inverse b′⁻¹(•) is termeda canonical link function. A link function serves to link a random orstochastic component of a model, a probability distribution of aresponse variable, to a systematic component of a model (e.g., a linearpredictor). For example, given E(Y)=q(v)=β₀+β₁x₁+ . . . +β_(n)x_(n),g(v) is a non-linear link function that links a random component, E(Y),to a systematic component, (β₀+β₁x₁+ . . . +β_(n)x_(n)). For traditionallinear models in which the random component consists of an assumptionthat a response variable follows the Normal distribution, a canonicallink function is an identity link. The identity link specifies that anexpected mean of the response variable is identical to the linearpredictor, rather than to a non-linear function of the linear predictor.From the formula (3), Var(Y)=φV(μ), where V(μ)=b″(θ)=b″(b′⁻¹(μ))≧0 is apositive-definite variance function. (A continuously differentiablefunction w is positive-definite on a neighborhood of an origin D, ifw(0)=0 and w(x)>0 for every non-zero x∈D. A function isnegative-definite if the inequality is reversed.) The inverse mean-valuemapping function θ=b′⁻¹(μ) can be used to write ED*(θ, φ) in (1) in theequivalent form ED(μ, φ), which is explicit in the two parameters μ andφ of primary interest.

Convolution property for the ED(μ, φ) Family

A convolution property of the exponential dispersion family yields arelationship for statistical parameters of a distribution of homogeneoussample aggregates from an underlying distribution ED(μ, φ). Theconvolution property includes, but is not limited to: (1)f*(g*h)=(f*g)*h, where f, g, h are functions, (2) (cf)*g=c(f*g)=f*(cg),where c is a constant, f and g are functions.

Consider n independent, identically distributed random variablesY_(i)∈ED(μ, φ_(i)), where φ_(i)=φ/ω_(i) for ω_(i)>0 and i is an index,so that each Y_(i) have a same mean, but different inverse dispersionweights (i.e., weights assigned to inverse dispersion parameters; if adispersion parameter is φ, then an inverse dispersion parameter is 1/φ).Then from the formula (2), the random variable

$Y = {{( {1/\omega} ){\sum\limits_{i = 1}^{n}{\omega_{i}Y_{i}\mspace{14mu}{with}\mspace{14mu}\omega}}} = {\sum\limits_{i = 1}^{n}\omega_{i}}}$produces the cumulant generating function

$\begin{matrix}{{{K_{Y}(t)} = {{\sum\limits_{i = 1}^{n}{K_{Y_{i}}( {\omega_{i}{t/\omega}} )}} = \frac{{b( {\theta + {( {\phi/\omega} )t}} )} - {b(\theta)}}{\phi/\omega}}},} & (4)\end{matrix}$so that a weighted sum

$( {1/\omega} ){\sum\limits_{i = 1}^{n}{\omega_{i}Y_{i}}}$has the same distribution Y∈ED(μ, φ/ω) as the individual Y_(i), withE(Y)=μ, Var(Y)=(φ/ω)Var(μ).

A case ω_(i)=1 in the formula (4) with Var(Y)=φ Var(μ)/n, where is thenumber of sample data, is of special interest, and the μ and φparameters for n identically distributed Y_(i)εED*(μ, φ) can beestimated from a distribution of an aggregate

${Y = {( {1/n} ){\sum\limits_{i = 1}^{n}Y_{i}}}},$since from the formula (4) Y belongs to ED(μ, φ/n). This result isfrequently used to fit exponential dispersion models (e.g., anexponential form of the GLM) to data in contingency tables, where thegroup averages in each cell k correspond to an aggregate distributionwith mean μ_(k) and dispersion φ_(k)/ω_(k), in which the weights ω_(k)are the known sample sizes of the aggregated data in cell k, and μ_(k)and φ_(k) are the respective mean and dispersion of an underlyingdistribution. The contingency tables are used to record and analyze arelationship between two or more variables, most usually categoricalvariables. Variables assessed on a nominal scale are called categoricalvariables.

For notational simplicity, φ_(k) has been used throughout to denote avariable dispersion parameter, but when known case weights ω_(k) areexplicitly required, as described above, then φ_(k) must be replaced byφ_(k)/ω_(k) wherever it appears.

Deviance and Negative Log-Likelihood Functions

For a random variable Y˜ED*(θ, φ) with a mean μ and variance functionV(μ)=Var(μ), a deviance function d(y, μ) is defined by

$\begin{matrix}{{d( {y,\mu} )} = {{- 2}{\int_{y}^{\mu}{\frac{y - s}{V(s)}\ {{\mathbb{d}s}.}}}}} & (5)\end{matrix}$

Thus, noting that s=b′(t), V(s)=b″(t),

$\begin{matrix}{{d( {y,\mu} )} = {{{- 2}{\int_{b^{\prime - 1}{(y)}}^{b^{\prime - 1}{(\mu)}}{( {y - {b^{\prime}(t)}} )\ {\mathbb{d}t}}}} = {- {{2\lbrack {{y\;\theta} - {b(\theta)} - {a(y)}} \rbrack}.}}}} & (6)\end{matrix}$Comparing (6) with (1),a(y)=yb′−1(y)−b(b′−1(y)),   (7)from which an identity a′(•)≡b′−1(•). From the formula (6), it followsthat

$\begin{matrix}{{\frac{\partial^{2}d}{\partial y^{2}} = {\frac{2}{V(\mu)} > 0}},} & (8)\end{matrix}$so that the d(y, μ) is convex (i.e., being a continuous function or partof a continuous function) as a function of y. A deviance refers to ameasure for judging an extent to which a model explains a variation in aset of data when the parameter estimation is carried out using a methodof maximum likelihood. An identity refers to an equality that remainstrue regardless of the values of any variables that appear in theformula (7).

Similarly,

$\begin{matrix}{{\frac{\partial^{2}d}{\partial\mu^{2}} = {2\{ {1 + {( {y - \mu} )\frac{V^{\prime}(\mu)}{V(\mu)}}} \}\frac{1}{V(\mu)}}},} & (9)\end{matrix}$so that d(y, μ) is monotonic (i.e., a monotonic function is a functionthat tends to move in only one direction as value(s) of parameter(s)increase), but is non-convex (i.e., being a non-continuous function orpart of non-continuous function) as a function of μ when μ−V(μ)/V′(μ)>y.In particular, for Tweedie distributions which are a subset of theexponential dispersion family with variance function V(μ)=μ^(p), thedeviance is non-convex for p>1 if μ(p−1)/p>y. However, an Expected Value

${E( \frac{\partial^{2}d}{\partial\mu^{2}} )} = {{2/{V(\mu)}} > 0}$of the formula (9) is always positive function of μ.

From the formula (1) and (6), the negative log-likelihood function l(y;μ, φ) for a random variable Y˜ED(μ, φ) can be written in the forml(y; μ,φ)=d(y, μ)/2φ/+c(y,φ).  (10)

Since principal minors of a Hessian of the formula (10) must be positivein order for l(y;μ, φ) to be convex, it follows from the discussionfollowing the formula (9), that l(y;μ,φ) is non-convex as a function ofμ and φ if either d(y, μ) is non-convex, or if φ>2d(y, μ). l(y; μ,φ) canbe non-convex even when the corresponding d(y, μ) is always convex (sofor example, l(y; μ,φ) is non-convex for the Normal distribution eventhough its deviance function d(y, μ) is always convex). Regions ofnon-convex geometry in the likelihood surface l(y; μ,φ) haveimplications for optimization algorithms used for joint modeling of amean and dispersion in the formula (4). if A is an m×n matrix, I is asubset of {1, . . . ,m} with k elements and J is a subset of {1, . . .,n} with k elements, then [A]_(I,J) for the k×k minor of A correspondsto rows with an index in I and columns with an index in J. If I=J, then[A]_(I,J) is called a principal minor. Hessian (matrix) is a squarematrix of second-order partial derivatives of a function. For example,given a real-valued function f(x₁, x₂, . . . , x_(n)), then the Hessianbecomes

${H(f)} = {\lbrack \begin{matrix}\frac{\partial^{2}f}{\partial x_{1}^{2}} & \frac{\partial^{2}f}{{\partial x_{1}}{\partial x_{2}}} & \ldots & \frac{\partial^{2}f}{{\partial x_{1}}{\partial x_{n}}} \\\frac{\partial^{2}f}{{\partial x_{2}}{\partial x_{1}}} & \frac{\partial^{2}f}{\partial^{2}x_{2}^{2}} & \ldots & \frac{\partial^{2}f}{{\partial x_{2}}{\partial x_{n}}} \\\vdots & \vdots & \ddots & \vdots \\\frac{\partial^{2}f}{{\partial x_{n}}{\partial x_{1}}} & \frac{\partial^{2}f}{{\partial x_{n}}{\partial x_{2}}} & \cdots & \frac{\partial^{2}f}{\partial^{2}x_{n}^{2}}\end{matrix} \rbrack.}$

The Normal, Gamma and Inverse Gaussian Distributions

The Normal, Gamma and Inverse Gaussian distributions are the onlymembers of the exponential distribution family for which c(y, φ) can bewritten in the form

$\begin{matrix}{{{c( {y,\phi} )} = {\frac{1}{2}\lbrack {{\hat{t}(y)} - {\hat{s}( \phi^{- 1} )}} \rbrack}},} & (11)\end{matrix}$where {circumflex over (t)}( ) is a specific function such as log 2π andŝ( ) is a specific function, e.g., log( ). Then, the negativelog-likelihood function (10) takes a special form

$\begin{matrix}{{l( {{y;\mu},\phi} )} = {{\frac{1}{2}\lbrack {{\phi^{- 1}{d( {y,\mu} )}} - {\hat{s}( \phi^{- 1} )} - {\hat{t}(y)}} \rbrack}.}} & (12)\end{matrix}$

This special form leads to certain simple iterative schemes for amaximum-likelihood estimation of the mean and dispersion parameters asdescribed below. The maximum-likelihood estimation (MLE) is a popularstatistical method used for fitting a statistical model to data, andproviding estimates for the model's parameters. For example, suppose auser is interested in heights of Americans. The user may have a sampleof some number of Americans, but not an entire population, and recordtheir heights. Further, the user is willing to assume that heights arenormally distributed with some unknown mean and variance. The samplemean is then the maximum likelihood estimation of the mean of thepopulation, and the sample variance is a close approximation to themaximum likelihood estimation of the variance of the population.

These distributions (i.e., Gamma, Normal and Inverse Gaussiandistributions) are also members of the Tweedie family for which thevariance function has a special form, V(μ)=μ^(p), where p is a parameterindicating a specific distribution. For the Normal (p=0), Gamma (p=2)and Inverse Gaussian (p=3) cases, respective cumulant functions b_(p)(θ)can be obtained by solving b″_(p)(θ)=(b′_(p)(θ))^(p) (and constants ofan integration can be set to convenient simplifying values). Similarly,respective deviance functions d_(p)(y, μ) can also be evaluated bysubstituting the corresponding variance function in the integral (5).These three cases are summarized below.

Normal Distribution, N(μ,σ)

The Normal distribution is a ED(μ, φ) distribution with φ=σ² for whichb(θ)=θ²/2,  (13)d(y, μ)=(y−μ)²,  (14)so that the canonical link function is the identity link. Then comparingthe negative likelihood with the formula (10),

$\begin{matrix}{{{a(y)} = {y^{2}/2}},{{c( {y,\phi} )} = {\frac{1}{2}\log\; 2{\pi\phi}}},} & (15)\end{matrix}$so that ŝ(φ⁻¹)=log φ⁻¹, and from the formula (15),∂c/∂φ=(½)φ⁻¹,∂²c/∂φ²=−(½)φ⁻².

Gamma Distribution, Ga(α,β)

The Gamma distribution is a ED(μ, φ) distribution with μ=α/β, φ=1/ α,for whichb(θ)=−log(−θ),  (16)d(y, μ)=2[y/μ−log(y/μ)−1],  (17)so that the canonical link function is the negative reciprocal link,e.g., −1/θ. From the formula (10),a(y)=−(1+log y)   (18)c(y, θ)=φ⁻¹−φ⁻¹ log φ⁻¹+log y+log Γ(φ⁻¹),  (19)where Γ is a Gamma function, i.e., an extension of a factorial functionto real and complex number, so that ŝ(φ⁻¹)=2[φ⁻¹ log φ⁻¹−φ⁻¹−logΓ(θ⁻¹)]. In this case (Gamma distribution), to leading order for φ→0,then c(y,φ)≈(½)log(2π log(2πφy²)+φ/12+O(φ³). However, c(y,φ) of theformula (19) and its required derivatives can be evaluated in terms ofwell-known special functions, e.g., well-known Digamma and well-knownTrigamma functions, as follows

$\begin{matrix}{{\frac{\partial c}{\partial\phi} = {\phi^{- 2}\lbrack {{\log( \phi^{- 1} )} - {\Psi( \phi^{- 1} )}} \rbrack}},} & (20) \\{{\frac{\partial^{2}c}{\partial\phi^{2}} = {\phi^{- 3}\lbrack {1 + {2{\log( \phi^{- 1} )}} - {2{\Psi( \phi^{- 1} )}} - {\phi^{- 1}{\Psi^{\prime}( \phi^{- 1} )}}} \rbrack}},} & (21)\end{matrix}$where Ψ(.) and Ψ′(.) denote the Digamma and Trigamma functionsrespectively.

Inverse Gaussian Distribution

The Inverse Gaussian distribution IG(μ, σ) is a ED(μ, φ) distributionwith φ=σ², so thatb(θ)=2 θ^(1/2),  (22)d(y, μ)=[(y/μ2)=(2/μ)+(1/y)],  (23)so that the canonical link is a square reciprocal link, e.g.,

$\frac{1}{\sqrt{\theta}}.$From the formula (10), a(y)=−y⁻¹,   (24)

$\begin{matrix}{{c( {y,\phi} )} = {\frac{1}{2}\log\; 2{\pi\phi}\;{y^{3}.}}} & (25)\end{matrix}$

Thus ŝ(φ⁻¹)=log φ⁻¹, and from the formula (25),∂c/∂φ=(½)φ⁻¹, ∂² c/∂φ²=−(½)φ⁻².

Relevance to Other Distributions

The form c(y, φ) in the formula (11), which is exact only for theNormal, Gamma and Inverse Gaussian distributions, also has the same formwith a saddlepoint density approximation in a leading-order term for φ→0for other conditional response distributions from the exponentialdispersion family. The saddlepoint density approximation is a formulafor approximating a density function from its associated cumulantgenerating function. The leading-order term is an approximation to agiven function, usually under assumptions about the inputs being verysmall or very large. The conditional response distribution is a set ofmeasurements reflecting distributional changes across sample data.

Consider an informal derivation of the saddlepoint approximation to thelikelihood for an exponential dispersion family distribution. Thedensity function f(y;θ,φ) of a random variable Y ˜ED*(θ,φ) can berepresented as an inverse Fourier transform of a characteristic functionwith imaginary argument exp {K_(Y)(iz)} based on the formula (4), sothat

$\begin{matrix}{{{f( {{y;\theta},\phi} )} = {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{\mathbb{e}}^{K_{Y}{({iz})}}\ {\mathbb{e}}^{- {izy}}{\mathbb{d}z}}}}},{{where}\mspace{14mu} z\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{parameter}},} & (26)\end{matrix}$

The (probability) density function is a function which describes adensity of probability each point in sample data.

For a density from the exponential dispersion family, from the formula(2), after setting {tilde over (θ)}=b′⁻¹(y), where {tilde over (θ)} isthe saddlepoint approximation to θ, noting from the formula (7) thata(y)=y{tilde over (θ)}−b({tilde over (θ)}), and then comparing theformula (26) with the formula (1), gives for c(y;φ) that

$\begin{matrix}{{c( {y;\phi} )} = {{- \log}\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{\mathbb{e}}^{{({{b{({\theta + {i\;\phi\; z}})}} - {b{(\overset{\sim}{\theta})}} - {y{({\theta + {i\;\phi\; z}})}} + {y\overset{\sim}{\theta}}})}/\phi}\ {\mathbb{d}z}}}}} & (27)\end{matrix}$

A change of variable iz=s takes a line of integration to an imaginaryaxis in complex plane for s. However, since an integrand in the formula(27) is analytic and vanishes for iz→±i∞, a value of an inversionintegral will be unaffected by translating the line of integrationparallel to the imaginary axis, in particular so that it passes througha saddlepoint of an argument of the exponential (the saddlepoint of afunction of two variables is defined as a point where a gradient of thefunction is zero, and the Hessian is indefinite, that is, the Hessianhas a positive and a negative eigenvalue). Thus, using the change ofvariables iz=s−(θ−{tilde over (θ)})/φ, so that

$\begin{matrix}{{c( {y;\phi} )} = {{- \log}\frac{1}{2\pi\; i}{\int_{{- i}\;\infty}^{{+ i}\;\infty}{{\mathbb{e}}^{({{b{({\overset{\sim}{\theta} + {\phi\; s}})}} - {b{(\overset{\sim}{\theta})}} - {{({\phi\;{sy}})}/\phi}}}\ {{\mathbb{d}s}.}}}}} & (28)\end{matrix}$

A steepest ascent method can now be used to evaluate the integral, whichfor leading order as φ→0 is dominated by the contribution of theintegrand in a neighborhood of the saddlepoint. The steepest ascentmethod is described in this context in “Explaining the SaddlepointApproximation”, C. Goutis and G. Casella, The American Statistician,Vol. 53(3), pp. 216-224, 1999, wholly incorporated by reference as ifset forth here in, and involves finding approximations to line integralsin situations where a parameter in an integrand becomes small, as in(28)). Thus

$\begin{matrix}{{c( {y;\phi} )} \approx {{- \log}\frac{1}{2\;\pi\; i}{\int_{{- i}\;\infty}^{{+ i}\;\infty}{{\mathbb{e}}^{\frac{1}{2}\phi\;{b^{''}{(\overset{\sim}{\theta})}}s^{2}}\ {\mathbb{d}s}}}}} \\{= {\log\sqrt{2\;{\pi\phi}\;{V(y)}}}} \\{{= {\overset{\sim}{c}( {y;\phi} )}},}\end{matrix}$in which {tilde over (c)}(y;φ) denotes the saddlepoint approximation toc(y; φ). Various steps in a formula (29) follow from a fact that b(θ) isat least twice-differentiable, and at a unique saddlepoint s=0 ofb({tilde over (θ)}+iφs)−b({tilde over (θ)})−iφs, b″({tilde over(θ)})=V(y)>0 (a special case when V(y)=0 is addressed below).

From the formula (29), the negative log-likelihood (10) with thisun-normalized saddlepoint approximation to the exact density can bewritten as

$\begin{matrix}{{l( {{y;\mu},\phi} )} \approx {\frac{d( {y,\mu} )}{2\;\phi} + {\log{\sqrt{2{\pi\phi}\;{V(y)}}.}}}} & (29)\end{matrix}$

Thus the formula (29) which is valid to leading order for φ→0, has asame form as the formula (12), with ŝ(φ⁻¹)≈−log φ.

In fact, this approximation (29) is also an exact negativelog-likelihood function for the Normal and Inverse Gaussiandistributions, and a renormalization of the formula (29) leads to anexact negative log-likelihood for the Gamma distribution as well. Thenegative log-likelihood is a negative log of a probability of anobserved response. Minimizing the negative a log-likelihood functionthus produces maximum likelihood estimates for a particular effect.

This saddlepoint likelihood (29) provides a computationally simpleralternative to an exact negative log-likelihood for joint estimation ofμ and φ in the small dispersion (φ→0) limit, and is equivalent to anextended quasi-likelihood function, that has been widely used in GLMmodeling (albeit for more general variance functions which are not justrestricted to the exponential dispersion family). The quasi-likelihoodfunction is a function which has similar properties to log-likelihoodfunction, except that a quasi-likelihood function is not thelog-likelihood corresponding to any actual probability distribution.

For Tweedie family distributions Tw_(p)(μ, φ) with 1≦p<2, i.e.,specifically for the Poisson and Compound-Poisson distributions, thevalue y=0 is in the range of the distribution with V(0)=0. Tweediedistributions are a family of probability distributions which includecontinuous distributions such as the Normal and Gamma, a purely discretescaled Poisson distribution, and a class of mixed compound Poisson-Gammadistributions.

Joint Regression Modeling

Loss Function for Joint Modeling

A loss function for joint regression modeling parameters in a regressionmodel are estimated by minimizing this loss function) is the empiricalnegative log-likelihood for a conditional response variable from theED(μ, φ) family over the training data records {y_(i), x_(i),z_(i)}_(i=1) ^(n). From the formula (10), this loss function is given by

$\begin{matrix}{{L( {\mu,\phi} )} = {{\sum\limits_{i = 1}^{n}\;{l( {{y_{i};\mu_{i}},\phi_{i}} )}} = {\sum\limits_{i = 1}^{n}{\lbrack {\frac{d( {y_{i},\mu_{i}} )}{2\;\phi_{i}} + {c( {y_{i},\phi_{i}} )}} \rbrack.}}}} & (30)\end{matrix}$

The regression functions for the mean and dispersion are given by η(x)and ζ(z) respectively, where x and z denote respective vector ofcovariates (these two sets of individual covariates need not be mutuallyexclusive). Then, given suitable invertible link functions, g:Range(Y)→IR and h: IR⁺→IR, theng(μ)=η(x), h(φ)=ζ(z).  (31)

For twice-differentiable loss (30) and the link (31) functions, requiredparameter estimation can be carried out using Newton iteration (i.e., amethod for finding successively better approximation to zeros (or roots)of a real-valued functions), which require following derivatives

$\begin{matrix}{{{\frac{\partial l_{i}}{\partial\eta} = {- \frac{( {y_{i} - \mu_{i}} )}{\phi_{i}{V( \mu_{i} )}{g^{\prime}( \mu_{i} )}}}},{\frac{\partial l_{i}}{\partial\zeta} = {{- ( {\frac{d_{i}}{2\;\phi_{i}^{2}} - \frac{\partial c_{i}}{\partial\phi_{i}}} )}\frac{1}{h^{\prime}( \phi_{i} )}}},{{{where}\mspace{14mu}{denoting}\mspace{14mu} d_{i}} = {{{d( {y_{i},\mu_{i}} )}\mspace{14mu}{and}\mspace{11mu} c_{i}} = {{c( {y_{i},\phi_{i}} )}.\mspace{14mu}{Similarly}}}},}\;} & (32) \\{{\frac{\partial^{2}l_{i}}{\partial\eta^{2}} = {\lbrack {1 + {( {y_{i} - \mu_{i}} )\{ {\frac{V^{\prime}( \mu_{i} )}{V( \mu_{i} )} + \frac{g^{''}( \mu_{i} )}{g^{\prime}( \mu_{i} )}} \}}} \rbrack\frac{1}{\phi_{i}{V( \mu_{i} )}( {g^{\prime}( \mu_{i} )} )^{2}}}},} & (33) \\{\frac{\partial l_{i}}{{\partial\eta}{\partial\zeta}} = {\frac{( {y_{i} - \mu_{i}} )}{\phi_{i}^{2}{V( \mu_{i} )}{g^{\prime}( \mu_{i} )}{h^{\prime}( \phi_{i} )}}.}} & (34) \\{\frac{\partial^{2}l_{i}}{\partial\zeta^{2}} = {\lbrack {( {\frac{d_{i}}{\phi_{i}^{3}} + \frac{\partial^{2}c_{i}}{\partial\phi_{i}^{2}}} ) + {( {\frac{d_{i}}{2\phi_{i}^{2}} - \frac{\partial c_{i}}{\partial\phi_{i}}} )\frac{h^{''}( \phi_{i} )}{h^{\prime}( \phi_{i} )}}} \rbrack{\frac{1}{( {h^{\prime}( \phi_{i} )} )^{2}}.}}} & (35)\end{matrix}$

From the formula (34), it can be seen the μand φ parameters for anexponential dispersion family distribution are statistically orthogonal,as E(∂²l/∂μ∂φ)=0.

A use of the canonical link g(μ)=b′⁻¹(μ) in mean regression model leadsto considerable simplification in the formulas (32)-(35), sinceg′(μ)V(μ)=1, g″(μ)V′(μ)+g′(μ)V′(μ)=0 in that case.

Joint Modeling Using GLM's

General Case

FIG. 1 illustrates a flow chart describing method steps for performingjoint modeling of a mean and dispersion using GLM. At step 100, thecomputing system 1600 obtains a loss function such as the formula (30).Alternatively, a user may provide the loss function to the computingsystem 1600. At step 110, the computing system 1600 represents aregression function for the mean as a linear function of covariate(i.e., an independent variable not manipulated by an experiment butstill affecting an outcome or response) and represents a regressionfunction for the dispersion as linear function of covariate. Accordingto one embodiment of the present invention, the GLM for mean regressionmodeling is extended to jointly modeling the mean and dispersion, e.g.,by taking η(x) and ζ(z) to be linear functions of covariates,η(x)=β^(T) x, ζ(z)=γ^(T) z,   (36)where β and γ are vectors of coefficients, Intercept terms, whichcorrespond to constant terms in this linear function, can beincorporated in the formula (36) without changing the notation in (36)by adding a constant-valued dummy covariate to the sets of covariates xand z respectively.

At step 120, the computing system 1600 estimates the covariates orcoefficients in the linear functions, e.g., using the Newton iteration.The gradient (∇) and Hessian (H) of the formula (30) with respect tomodel coefficients in (36) is given by

$\begin{matrix}{{( {\nabla_{\beta}l} )_{k} = {\sum\limits_{i = 1}^{n}{( \frac{\partial l_{i}}{\partial\eta} )_{k}x_{i}}}},{( {\nabla_{\gamma}l} )_{k} = {\sum\limits_{i = 1}^{n}{( \frac{\partial l_{i}}{\partial\zeta} )_{k}z_{i}}}},{and}} & (37) \\{{( H_{\beta\beta} )_{k} = {\sum\limits_{i = 1}^{n}\;{( \frac{\partial^{2}l_{i}}{\partial\eta^{2}} )x_{i}x_{i}^{T}}}},{( H_{\gamma\gamma} )_{k} = {\sum\limits_{i = 1}^{n}{( \;\frac{\partial^{2}l_{i}}{\partial\zeta^{2}} )_{k}z_{i}z_{i}^{T}}}},} & (38) \\{{( H_{\beta\gamma} )_{k} = {\sum\limits_{i = 1}^{n}{( \frac{\partial^{2}l_{i}}{{\partial\eta}{\partial\zeta}} )_{k}x_{i}z_{i}^{T}}}},} & (39)\end{matrix}$

These derivatives, when taken in conjunction with the formulas(32)-(35), lead to the following Newton iteration for estimating {β,γ}in the form

$\begin{matrix}{\begin{bmatrix}\beta_{k + 1} \\\gamma_{k + 1}\end{bmatrix} = {\lbrack \;\begin{matrix}\beta_{k} \\\gamma_{k}\end{matrix} \rbrack - {\begin{bmatrix}( H_{\beta\beta} )_{k} & ( H_{\beta\gamma} )_{k} \\( H_{\gamma\beta} )_{k} & ( H_{\gamma\gamma} )_{k}\end{bmatrix}^{- 1}\begin{bmatrix}( {\bigtriangledown_{\beta}l} )_{k} \\( {\bigtriangledown_{\gamma}l} )_{k}\end{bmatrix}}}} & (40)\end{matrix}$

Alternatively, the computing system 1600 utilizes a Fisher scoringalgorithm for GLM, in which the Hessian matrix (38)-(40) is replaced byits Expectation (an approximation that becomes increasingly moreaccurate near an optimum solution). From the formula (34), it can beseen that a cross-derivative term in the formula (39) has an Expectationof 0, so that the Fisher scoring algorithm effectively decouples theupdating of the parameters for the mean and dispersion, as follows,β_(k+1)=β_(k) −[E((H _(ββ)))_(k)]⁻¹(∇_(β) l)_(k),   (41)γ_(k+1)=γ_(k) −[E((H _(γγ)))_(k)]⁻¹(∇_(γ) l)_(k).   (42)

Since E((H_(ββ)) in the formula (41) is always positive-definite, theteration (41)-(42) will be globally convergent when E((H_(γγ))) is alsopositive-definite. The Fischer scoring algorithm is a form of Newtoniteration used to solve maximum likelihood equations numerically. Themaximum likelihood equations are statistical equations used for fittinga statistical model to sample data.

After completing the estimations of the covariates or coefficients,e.g., the formulas (40)-(42), in the linear regression functions (e.g.,the formula 36), the computing system 1600 completes the linearregression functions (36) with the estimated covariates or coefficients.These regression functions (36) approximate the mean and dispersion ofsample data concurrently and/or jointly.

Special Case

Consider the formulas (41)-(42) for the case of the Normal, Gamma andInverse Gaussian distributions, where the negative log-likelihood takesthe special form (12).

From the formulas (32) and (33), in this case of the Normal, Gamma andInverse Gaussian distributions,

$\begin{matrix}{{{E( y_{i} )} = \mu_{i}},{{{E( d_{i} )} \equiv \delta_{i}} = {\overset{\overset{\prime}{\hat{}}}{s}( \phi_{i}^{- 1} )}},} & (43)\end{matrix}$where ′ represents a first-order differential.If denoting training data byy=[y₁, . . . , y_(n)]^(T), X=[x₁, . . . , x_(n)]^(T), Z=[z₁, . . . ,z_(n)]^(T),  (44)From the formula (36),g(μ)=β^(T) X, h(φ)=γ^(T) Z.  (45)From the formulas (33)-(35), define diagonal matrices

$\begin{matrix}{{W = {{Diag}\{ {{\phi_{i}^{- 1}( {V( \mu_{i} )} )}^{- 1}( {g^{\prime}( \mu_{i} )} )^{- 2}} \}}},} & (46) \\{{V = {{Diag}\{ {\begin{matrix} {{- \frac{1}{2}}\phi_{i}^{- 4}{\overset{\overset{''}{\hat{}}}{s}( \phi_{i}^{- 1} )}} )\end{matrix}( {h^{\prime}( \phi_{i} )} )^{- 2}} \}}},} & (47) \\{{D = {{Diag}\begin{Bmatrix} {{- \phi_{i}^{- 2}}{\overset{\overset{''}{\hat{}}}{s}( \phi_{i}^{- 1} )}} )\end{Bmatrix}}},} & (48)\end{matrix}$where ″ represents a second-order differentialand let

$\begin{matrix}{d = \lbrack {{d( {( {\mu_{1},y_{1}} ),\ldots\mspace{14mu},{d( ( {\mu_{n},y_{n}} ) \rbrack}^{T},{\delta = \lbrack {{\overset{\overset{\prime}{\hat{}}}{s}( \phi_{1}^{- 1} )},\ldots\mspace{14mu},{\overset{\overset{\prime}{\hat{}}}{s}( \phi_{n}^{- 1} )}} )}} \rbrack}^{T}.} } & (49)\end{matrix}$Thus, denoting working responses at an iteration k by{circumflex over (μ)}_(k)=g′(μ_(k))(y−μ _(k))+g(μ_(k)),  (50)φ_(k)=h′(φ_(k))D _(k) ⁻¹({circumflex over(d)}_(k)−δ_(k))+h(φ_(k)),  (51)the iterations in the formulas (41) and (42) can be written in a form,β_(k+1)=(X ^(T) W _(k) X)⁻¹ X ^(T) W _(k){circumflex over (μ)}_(k),  (52)γ_(k+1)=(Z ^(T) V _(k) Z)⁻¹ Z ^(T) V _(k){circumflex over (φ)}_(k).  (53)

Each iteration step in the formulas (52)-(53) then corresponds to a pairof weighted least squares problems (i.e., a variant of a standard leastsquares problem in which each case is weighted differently whichtypically arises iterative methods; “Iteratively Reweighted LeastSquares for Maximum Likelihood Estimation, and some Robust and ResistantAlternatives,” P. J. Green, Journal of the Royal Statistical Society,Volume 46(2), pages 149-192, 1984, wholly incorporated by reference asif set forth herein, describes the weighted least square problems in aniterative setting in detail.), in which the working responses (50)-(51),and the weight vectors W_(k), V_(k) in (48) are updated at eachiteration.

An overall convergence of the iteration can be improved by using themost recently updated values μ_(k+1), from a half-step (52) (i.e., halfprocessing of the formula (52)) to obtain the working response andweight vector in order to solve for φ_(k+1) from a half-step (53). Inthis way, the iteration alternates between the formulas (52) and (53),always using the most recent updates in each half-step of the iteration.

The formula (53) used to update the dispersion parameter sub-model(i.e., a formula or model calculating the dispersion parameter) can beinterpreted as GLM model in its own right, since the negativelog-likelihood in the formula (12) can be regarded as a“pseudo-deviance” which has a same form as the formula (6), afterinterpreting a deviance residual d as a response, φ⁻¹ as the canonicalparameter, and ŝ(φ⁻¹) as the cumulant function respectively.

The Gamma distribution (described supra) is the only case where theformula (12) holds with a non-trivial form for ŝ(φ⁻¹), with the formula(19)

$\begin{matrix}{{{\overset{\overset{\prime}{\hat{}}}{s}( \phi_{i}^{- 1} )} = {2\lbrack {{\log\;\phi_{i}^{- 1}} - {\Psi( \phi_{i}^{- 1} )}} \rbrack}},{{\overset{\overset{''}{\hat{}}}{s}( \phi_{i}^{- 1} )} = {{2\lbrack {\phi_{i} - {\Psi^{\prime}( \phi_{i}^{- 1} )}} \rbrack}.}}} & (54)\end{matrix}$

In the case of the Normal and Inverse Gaussian distributions, ŝ(φ⁻¹)=logφ_(i) ⁻¹. Then from the formula (43), E(d_(i))=φ_(i) and

${{{- \phi_{i}^{- 2}}{\overset{\overset{''}{\hat{}}}{s}( \phi^{- 1} )}} = 1},$so that in this case D_(k) in the formula (51 ) can be set to theidentity matrix, with other simplifications in V_(k) as well. Therefore,an iteration for the dispersion sub-model is equivalent to the Fisherscoring algorithm for a response d_(i) using a GLM for whichE(d_(i))=φ_(i), the Gamma variance function V(φ_(i))=φ_(i) ², and with afixed dispersion parameter with value 2.

Alternatively, the special form (e.g., the formula (12) of thelikelihood function in these cases (i.e., the Normal, Gamma and InverseGaussian distributions) suggests that existing GLM macros (i.e., a ruleor pattern that specifies how a certain input sequence should be mappedto an output sequence according to GLM) can be used to handle jointmodeling of the mean and dispersion. The mean and dispersion sub-modelsin the formulas (52)-(53) are equivalent to loss functions used in theGLM. Therefore, starting from an initial estimate of φ, the computingsystem 1600 alternates between solving a GLM model for μ with φ heldfixed, followed by the GLM for φ with μ held fixed (the latter GLM willuse standard macros that are available for performing Gamma regressionin the Normal and Inverse Gaussian cases). Although this decoupledapproach simplifies programming by making use of existing GLM macros instatistical packages, a convergence is likely to be slower than thetightly-coupled iterations used in the formulas (52) and (53).

Joint Modeling of Mean Using MART

The MART methodology for regression modeling can also be extended forjoint modeling applications (i.e., joint modeling of a mean anddispersion) as follows. Let η₀ and ζ₀ denote constant offsets, anddenoting the sample response mean by y, on possible set of defaultvalues for constant offsets is η₀=g( y), ζ₀=h((1/(n−1))Σ_(i)d_(i)(y_(i),y)), where n is an arbitrary variable. These values (η₀ and ζ₀)correspond to the maximum saddlepoint likelihood estimates forunconditional response.

FIG. 2 illustrates a flow chart describing method steps for jointmodeling of a mean and dispersion by extending the MART methodologyaccording to one embodiment of the present invention. At step 200, thecomputing system 1600 obtains a deviance loss function (e.g., a formula(30)). In one embodiment, a user may provide the deviance loss functionto the computing system 1600. At step 210, the computing system 1600represents regression functions for the mean and dispersion as astagewise expansions, (expansions are functions that consists of sums ofindividual terms, and stagewise expansions are expansions whose termsare computed one by one, instead of all the terms simultaneously as is acase in many standard regression algorithms), which include theundetermined scalar coefficients and undetermined basis functions foreach term. In the required extension to the MART methodology, theregression functions in the formula (31) are given by the stagewiseexpansion forms:g(μ_(k))=η_(k)(x), h(φ_(k))=ζ_(k)(z),  (55)where, starting with offsets η₀ and ζ₀, at each stage, the computingsystem 1600 computesη_(k+1)(x)=η_(k)(x)+a _(k+1) R(x, β _(k+1)), ζ_(k+1)(z)=ζ_(k)(z)+b_(k+1) S(z, γ _(k+1)),  (56)In each stage, positive scalar coefficients {a_(k+1),b_(k+1)}, andparameterized basis functions R(x, β),S(z, γ) are chosen so as to obtaina significant reduction in the loss function (30).

However, a simultaneous optimization of all these variables in theformula (56) is computationally challenging, and thus the followingtwo-step algorithm is therefore used to develop a solution (i.e., acomplete regression stage function with a determined scalar coefficientsand a determined basis functions) at each stage. The basis functionrefers to an element of a particular basis (i.e., a set of vectors) fora function space (i.e., a set of functions). Every function in thefunction space can be represented as a linear combination of basisfunctions, just as every vector in a vector space can be represented asa linear combination of basis vectors.

Determination of the Stage Basis Functions

Returning to FIG. 2, at step 220, the computing system 1600 chooses thebasis functions that maximally correlate with a correspondingsteepest-descent gradient direction (i.e., a gradient method in which achoice of a direction is where a function f decreases most quickly,which is the direction opposite to a gradient of the function f) of thedeviance loss function. At first, the parameters {β_(k+1), γ_(k+1)} foroptimal basis functions in the formula (56) are estimated from

$\begin{matrix}{{\beta_{k + 1} = {\arg\;{\min_{\beta}{\sum\limits_{i = 1}^{n}\lbrack {( {- \frac{\partial l_{i}}{\partial\eta}} )_{k} - {R( {x_{i},\beta} )}} \rbrack^{2}}}}},} & (57) \\{\gamma_{k + 1} = {\arg\;{\min_{\gamma}{\sum\limits_{i = 1}^{n}{\lbrack {( {- \frac{\partial l_{i}}{\partial\zeta}} )_{k} - {R( {z_{i},\gamma} )}} \rbrack^{2}.}}}}} & (58)\end{matrix}$where “arg min” denotes a value of an argument (e.g., β in (57)) forwhich the function has the minimum value. As a result of (57) and (58),the basis functions {R(x, β_(k+1)), S(z, γ_(k+1))} in the formula (56)are therefore chosen so as to be maximally correlated with acorresponding steepest-descent gradient direction of the loss function(30) evaluated at the previous stage in the expansion (56). In MARTmethodology, as described in Friedman, these basis functions areregression trees (i.e., function approximations in which a response ispredicted over individual domains obtained by partitioning a space ofcovariate features) of fixed depth, computed using least-squaressplitting criterion (57) and (58) (the least-squares splitting criterionis used to subdivide a given domain in a regression tree into tworegions in a recursive manner, if in doing so, a mean-square deviationof a response in the domain is reduced), so that the parameters{β_(k+1), γ_(k+1)} represent covariate split conditions (i.e., aconcatenation of all the conditions on a covariate that are used for adomain subdivision in the regression tree) and leaf node responseestimates (i.e., mean values of a response in each subdomain in a finalpartitioning of a regression tree) in the respective regression trees.The use of regression trees obtained with the least squares criterion asthe basis functions in the formula (56), is responsible for manyattractive computational properties of the MART methodology, as well asfor the present invention, as described further below.

The regression trees, which are chosen to be of small depth, typicallyeither 1 or 2, are termed “weak learners”. In particular, the use ofregression trees of depth 1 corresponds to piecewise-constant, additivenonlinear regression functions for η(x) and ζ(z). Similarly, the use ofregression trees of depth larger than 1 (e.g., depth 2) correspond topiecewise-constant, additive nonlinear regression functions η(x) andζ(z), that incorporate first-order local interaction terms(“first-order” implying that these terms involve simple products ofcovariates, and “local” implying that these terms are used only aparticular partition of the input covariate space, rather than globallythroughout an input covariate space). The regression trees used in theregression functions R(x, β) and S(z, γ) can have different depths,which can provide additional modeling flexibility (for example, for theconstant dispersion case, a tree of depth zero will suffice for S(z,γ)). A function is said to be piecewise-constant if it is locallyconstant in connected regions separated by a possibly infinite number oflower-dimensional boundaries. An additive function is a function thatpreserves an addition operation:ƒ(x+y)=ƒ(x)+ƒ(y)for any two elements x and y in a domain. A regression functiondescribes a relationship between a dependent variable y and explanatoryvariable(s) x.

Determination of the Expansion Coefficients

Returning to FIG. 2, at step 230, the computing system 1600 obtains thescalar coefficients, e.g., by running a single step of the NewtonIteration. After the R(x, β_(k+1)) and S(z, γ_(k+1)) have been computedfrom the formulas (57) and (58) as outlined above, the scalars{a_(k+1),b_(k+1)} are obtained from(a _(k+1) ,b _(k+1))=arg min_((a,b)) {circumflex over (L)}_(k)(a,b),  (59)where{circumflex over (L)} _(k)(a,b)=L(g ⁻¹(η_(k) +aR(x,β _(k+1))), h⁻¹(ζ_(k) +bS(z,γ _(k+1)))).  (60)

The formulas (59)-(60) represents a bivariate optimization problem,since it involves finding optimum values of two variates a and b, sothat the function {circumflex over (L)}_(k)(a,b) in (59)-(60) isminimized. Thus, a use of the Newton iteration is preferred since it isunaffected by a local geometry of an optimization surface (59) (i.e., ageometry of {circumflex over (L)}_(k)(a,b) as a function of (a,b)),since this geometry can be quite poorly scaled in an initial stages of astagewise procedure (56), whereas gradient line-search methods, in whichvalues of (a,b) in a direction of a gradient of the function {circumflexover (L)}_(k)(a,b) are examined to find optimum values, is sensitive toa relative scaling of optimization variables. A coupled optimization ofthe formula (59) for the variables a_(k+1) and b_(k+1) may be used, andparticularly in a vicinity of an optimum where the Newton iteration hasa rapid quadratic convergence. Quadratic convergence means that theerror at a given iteration is proportional to the square of an error atthe previous iteration. For example if an error is one significant digitat a given iteration, then at a next iteration it is two digits, thenfour, etc. After choosing the basis function and obtaining the scalarcoefficients, the computing system 1600 completes the regressionfunctions for the mean and dispersion based on the chosen basis functionand the obtained scalar coefficients.

An important aspect of using the Newton iteration is that starting froman initial guess (a, b)=(0, 0), only a single Newton step is taken,rather than iterating the optimization (59) to convergence. This “earlystopping” helps in avoiding over-fitting (over-fitting is a property ofa regression function, when it fits the sample data with little or noerror; such regression functions typically do very poorly with makingpredictions on unseen data, and these functions are also very sensitiveto new sample data which can lead to drastic changes in the form of theregression function), to specific basis functions that appear in earlystages of the stage-wise procedure (56), but without loss of overallaccuracy, since any sub-optimal solutions for the formula (59) at agiven stage will be corrected in subsequent stages of the overallstage-wise procedure (56). The single step of the Newton iteration tothe formula (59) only requires an evaluation of derivatives of theformula (59) at (a, b) (0, 0),

$\begin{matrix}{{( {\nabla_{a}\hat{L}} )_{k} = {\sum\limits_{i = 1}^{n}{( \frac{\partial l_{i}}{\partial\eta} )_{k}{R( {x_{i},\beta_{k + 1}} )}}}},{( {\nabla_{b}\hat{L}} )_{k} = {\sum\limits_{i = 1}^{n}{( \frac{\partial l_{i}}{\partial\zeta} )_{k}{S( {z_{i},\gamma_{k + 1}} )}}}},} & (61) \\{{( {\hat{H}}_{aa} )_{k} = {\sum\limits_{i = 1}^{n}{( \frac{\partial^{2}l_{i}}{\partial\eta^{2}} )_{k}{R( {x_{i},\beta_{k + 1}} )}^{2}}}},{( {\hat{H}}_{bb} )_{k} = {\sum\limits_{i = 1}^{n}{( \frac{\partial^{2}l_{i}}{\partial\zeta^{2}} )_{k}{S( {z_{i},\gamma_{k + 1}} )}^{2}}}},} & (62) \\{{( {\hat{H}}_{ab} )_{k} = {\sum\limits_{i = 1}^{n}{( \frac{\partial^{2}l_{i}}{{\partial\eta}{\partial\zeta}} )_{k}{R( {x_{i},\beta_{k + 1}} )}{S( {z_{i},\gamma_{k + 1}} )}}}},} & (63)\end{matrix}$which should be used in conjunction with the formulas (32)-(35), so that

$\begin{matrix}{\begin{bmatrix}a_{k + 1} \\b_{k + 1}\end{bmatrix} = {{- \hat{\alpha}} - {{\begin{bmatrix}{( {\hat{H}}_{aa} )_{k} + \hat{\lambda}} & ( {\hat{H}}_{ab} )_{k} \\( {\hat{H}}_{ba} )_{k} & {( {\hat{H}}_{bb} )_{k} + \hat{\lambda}}\end{bmatrix}^{- 1}\begin{bmatrix}( {\nabla_{a}\hat{L}} )_{k} \\( {\nabla_{b}\hat{L}} )_{k}\end{bmatrix}}.}}} & (64)\end{matrix}$

A choice {circumflex over (α)}=1 and {circumflex over (λ)}=0 in theformula (64) yields a usual single step of the Newton iteration for theformula (59). However, particularly in initial stages of the stagewiseprocedure, an optimization geometry may be non-convex and the Hessianmatrix in the formula (64) may not be positive-definite. Furthermore,even when the Hessian is positive-definite, the unmodified Newton stepcan occasionally lead to an increase rather than a desired decrease inan objective function (59). The parameters {circumflex over (α)} and βtherefore incorporate two simple modifications to an usual Newton step(i.e., a single step of the Newton iteration) to ensure globalconvergence in nonlinear optimization.

Returning to FIG. 2, at step 240, the computing system 1600 performs afirst modification. At the first modification, the computing system 1600modifies parameters (e.g., {circumflex over (α)} and β) used forobtaining the scalar coefficients to let the single step of the Newtoniteration be a steep-descent gradient direction for the obtained scalarcoefficients. The first modification requires computing the eigenvaluesof the 2×2 Hessian matrix in the formula (63), and in a semi-definite orindefinite case, the computing system 1600 sets {circumflex over(λ)}=−(1+δ){circumflex over (λ)}_(min), where {circumflex over(λ)}_(min)≦0 is the smallest eigenvalue, and δ>0 is a small positiveconstant, e.g., less than 10. This ensures that a modified Newtoniteration matrix is positive-definite and well-conditioned, so that amodified Newton step is indeed a descent direction for the objectivefunction (59).

Returning to FIG. 2, at step 250, the computing system 1600 performs asecond modification. At the second modification, the computing system1600 modifies parameters used for obtaining the scalar coefficients toachieve a decrease in values of the obtained scalar coefficients. Thesecond modification involves selecting 0<{circumflex over (α)}≦1 toensure that a size of a step in the descent direction from the modifiedNewton step leads to an actual decrease in the objective function (59),which is only guaranteed for the small enough {circumflex over (α)}.Therefore, starting with an initial value of {circumflex over (α)}=1 inthe formula (64) which would be an unmodified Newton step, the objectivefunction (59) evaluated with this step and if the objective function isnot sufficiently decreased according to a condition{circumflex over (L)} _(k)({circumflex over (α)}a _(k+1) ,{circumflexover (α)}b _(k+1))≦{circumflex over (L)} _(k)(0,0)−c ₁{circumflex over(α)}(a _(k+1)(∇_(a) {circumflex over (L)})_(k) +b _(k+1)(∇_(b){circumflex over (L)})_(k)),  (65)then the value of {circumflex over (α)} is successively halved till adesired condition (i.e., the obtained scalar coefficients reaches adesirable values) is satisfied (the constant value c1 in (65) mustsatisfy 0<c₁<1, e.g., c₁=10⁻⁴). This choice of {circumflex over (α)}ensures that the largest possible modified Newton step is taken thatconsistent with a sufficient decrease in the objective function (59) inthis stage Then, based on the modified parameters (e.g., modified{circumflex over (α)} and β), the computing system 1600 recalculates thescalar coefficients based on the formula (64). Then, the computingsystem 1600 obtains complete regression functions for the mean anddispersion based on the chosen basis function and the recalculatedscalar coefficients.

Optimum Number of Stages in Expansion

An optimum number of stages in the expansion (56), denoted by K, isobtained from the cross-validation estimate of the loss

$\begin{matrix}{{K = {\arg\;{\min_{k}{\sum\limits_{I = 1}^{N_{CV}}{L_{\{{- I}\}}( {\mu_{k}^{(I)},\phi_{k}^{(I)}} )}}}}},} & (66)\end{matrix}$

Where μ_(k) ^({I}), φ_(k) ^({I}) denote the mean and dispersionestimates respectively at the k'th stage from training data for the I'thfold, and L_({−I})(μ_(k) ^({I}), φ_(k) ^({I})) denotes a loss functionfor test data in the I'th fold at the k'th stage. The number ofcross-validation folds N_(CV) is typically 5 or 10. Other criteria suchas the 1−SE rule, in which the K is the smallest number of stages forwhich the cross-validation loss is within I standard error of a minimumcross-validation loss, can also be used as an alternative to the formula(66).

MART Implementation

For mean regression modeling, MART methodology leads to competitive,highly robust and interpretable procedures, and many of its advantagesreadily extend to the joint modeling of the mean and dispersiondiscussed supra. In particular, the use of basis functions comprised ofregression trees, which is fitted using the least squares criterion toresponses comprised of the steepest-descent gradient directions of thestagewise loss function, provides a systematic and modular approach to astagewise computation, that is independent of a specific form of anoverall loss function.

In particular, the use of the least-squares fitting criterion for theregression trees is computationally desirable, since the this criterioncan rapidly estimated using updating procedures while searching througha sequence of possible tree splits for an optimal split.

Another important benefit of the use of the least-squares fittingcriterion arises in a treatment of categorical covariate splits, where aconvexity of the least-squares splitting criterion ensures that if Ω isa cardinality of a categorical covariate, then the best split in thiscovariate can be found in just O(Ω) steps, without having to searchthrough the space of all possible splits which would require O(2^(Ω))steps. An ability to evaluate categorical splits in a linear rather thanexponential number of steps allows categorical features of highcardinality to be used in the regression modeling without requiring anypreprocessing and/or grouping of category levels in these features forcomputational tractability in a fitting procedure. In contrast,regression trees that directly use the overall loss function as thesplitting criterion may not have this useful property (i.e., notrequiring preprocessing and grouping) for categorical covariates, fromthe formula (9) relevant loss functions may not always be convex in amean regression, and are certainly non-convex in the joint regression ofthe mean and dispersion.

The use of regression trees as basis functions in the stagewiseexpansion is useful for high-dimensional data sets with numerous inputcovariates. Since an intrinsic dimensionality of the regression responsesurface is likely to be much smaller than the actual dimension of theinput covariate space, a greedy feature selection that is used in theregression trees ensures that collinear and irrelevant covariates areexcluded from regression models, so that there is no need to explicitlyincorporate a separate feature selection step in a computationalprocedure.

An ability of the regression functions to model low-order interactioneffects in the covariates can be achieved by using regression trees withdepth greater than 1, for example depth 2 or 3, since in practice, it isthe low-order interaction effects are typically the most important inthe regression modeling. However, from a model interpretation point ofview, the use of regression trees of depth of 1 has an advantage that ityields a regression function that is additive in the effects ofindividual covariates, so that it is possible to separate out effects ofindividual covariates in a final model.

Numerical Studies

It is difficult to find non-trivial data sets for a detailed study ofdistributional and heteroscedasticity effects in regression modeling.For most real-word data sets, many competing regression models providereasonable fits and consistent interpretations, and therefore in orderto illustrate relevant algorithmic issues, one embodiment of the presentinvention has also considered a few synthetic data sets with knownresponse characteristics in order to explicate some of the computationalresults presented below.

Synthetic Data Sets

A response variable in this collection of synthetic data sets is alwaysgenerated from a specific distribution with given mean and dispersiondependencies on the covariates. The present invention can then beevaluated in terms of both identifying a correct response model, andrecovering a known signal from the data.

The synthetic data sets in one implementation comprises of 1,000 sampleseach for training and validation. A covariate set is 6-dimensional,x={x₁, x₂, . . . , x₆}, where x₁, x₂, x₃ are continuous-valued anduniformly sampled in an interval (0, 1), and x₃, x₄, x₅ arecategorical-valued with 4 levels denoted {1, 2, 3, 4} respectively whichare sampled with equal probability. A response is given byy=Tw_(p)(μ(x), φ(x)) where p=0, 2, 3, Tw denotes the Tweediedistribution, and the mean and dispersion are given functions of thecovariates. In these data sets of this example implementation, the meanis an additive, nonlinear function of a form:μ(x)=7+10(x ₁−0.5)² −x+sin 2πx ₂−3x ₃+1.5I(x ₄).  (67)where I denotes an indicator function described in “Adaptivenonparametric estimation of mean and variance functions”, D. Chan, etal., Computational and Graphical Statistics, Vol. 15(4), pp. 915-936,2006, wholly incorporated by reference as if set forth herein, hereafter“Chan”.

A first collection of synthetic data sets, termed SYNTH1, are generatedwith a constant dispersion φ(x)=c for a response, with c being chosen toachieve a signal-to-noise ratio between 1 and 2 (for the Normaldistribution, a simulated response is checked to ensure strictlypositive values, so that non-normal loss functions can be used for modelexploration and comparison).

TABLE 1 Validation set loss estimate L_(v) for SYNTH1 data set, withregression tree depths l = 1 for the mean and k = 0 for the dispersion.The estimated standard errors are negligible and have been omitted forbrevity. Loss Function Response Normal Gamma Inverse Gaussian N (μ(x),0.6) 1.290 1.376 1.575 Ga (μ(x), 0.08) 2.100 2.000 2.042 Ga (μ(x), 1.0)3.386 2.895 7.749 IG (μ(x), 0.005) 1.728 1.604 1.628 IG (μ(x), 0.2)3.661 2.916 2.789

For example, in the Table 1, a response is generated from the Normal,Gamma and Inverse Gaussian distributions, and is fitted using all threeloss models. In all cases, the model fits use an identity link for themean sub-model (i.e., a regression function of the mean) and a log-linkfor the dispersion sub-model (i.e., a regression function of thedispersion), Results indicate that a “correct” loss model typicallyyields the lowest validation set loss estimate, and an examination ofthe model fit indicates a consistent signal recovery in each case. Forsmall c, an identification of the correct model can be confounded (ascan be noticed in the case of Gamma and Inverse Gaussian distributions)in the Table (1), but for larger c, a correct loss model also providesthe best model fit. A link function is a function that links a modelspecified in a design matrix, where columns represent beta parametersand rows represents real parameters. Then, the corresponding log linkfunction may be real parameters=exp(X*beta parameters), where X is arandom variable.

These results illustrated in the Table 1 also demonstrate a utility ofthe joint modeling, even in a constant dispersion case, sincealternative models for the conditional response can be compared, whichis not the case if only the deviance loss function were used.

TABLE 2 Validation set loss estimate L_(v) for SYNTH1 data set for theresponse given by N(μ(x), 0.6) for varying regression tree sizes (l, k).Stump depths (l, k) (1, 0) (1, 1) (2, 0) (2, 1) (2, 2) L_(V) 1.317 1.3931.386 2.588 69.7

The table 2 considers the data set for Normal-distributed response casein conjunction with the Normal loss function (i.e., a loss function in acase of Normal distribution) for a model fit, and describes an effect ofvarying regression tree sizes in the mean and dispersion sub-models. Thebest model accuracy is obtained with trees of depth (1, 0), which arethe simplest basis functions consistent with the assumed covanateeffects (i.e., dependence of an assumed regression function oncovariates in the data set) in the synthetic data. A use of more complexbasis functions in this case (i.e., Normal distribution case), leads toover-fitting effects in individual stages, with an overall sub-optimalsolution for a gradient boosting expansion (i.e., a machine learningalgorithm for fitting a weighted additive expansion of simple trees totraining data).

A second collection of data sets, termed SYNTH2, uses data generated fora variable dispersion case with a dispersion sub-model of a formφ(x)=c ₁ I ₁(x ₄)+c ₂ I ₂(x ₄).  (68)where I₁ is a first indicator function described in “Modeling VarianceHeterogeneity in Normal Regression Using GLM”, App. Statist, Vol. 36(3),pp. 332-339 (1987), wholly incorporated by reference as if set forthherein and I₂ is a second indicator function described in Chan.

TABLE 3 Validation set loss estimate L_(V) for SYNTH2 data set fittedwith a consistent loss model for a simulated response with results forvarying tree depths (l, k). Estimated standard errors are negligible andexcluded for brevity. Normal L_(V) (c₁ = 0.2, Gamma Inverse GaussianStump depths c₂ = 2.0) (c₁ = 0.08, c₂= 1.0) (c₁ = 0.08, c₂ = 0.8) l = 1,k = 0 1.598 2.822 2.810 l = 1, k = 1 1.421 2.566 2.655 l = 2, k = 01.661 2.876 2.817 l = 2, k = 1 1.490 2.597 2.634 l = 2, k = 2 1.5872.596 2.660

The table 3 describes for various response distributions, an effect ofvarying tree depths in respective fitted models in a variable dispersioncase, when a “correct” loss function is used for a model fit. For allthree response distributions (i.e., Normal, Gamma and Inverse Gaussian),a choice k=1 yields the best model fit, and this is also the simplestbasis function that is consistent with an assumed piecewise-constantvariation in the synthetic data.

Sniffer Data Set

A Sniffer data set, described in “Generalized Linear Models With VaryingDispersion”, G. K. Smyth, et al., Royal Statist. Soc. B., Vol. 51(1),pp. 47-60, 1989, wholly incorporated by reference as if set forthherein, can be used to model an amount y of hydrocarbon vapors escapingfrom a gasoline tank during filling, using four continuous explanatoryvariables, viz., respective temperatures (x₁ and x₂) and vapor pressures(x₃ and X₄) of an original and dispensed gasoline in the tank. This dataset, which comprises of 125 measurement records, can be obtained fromhttp://www.statsci.org/data/general/gasvapor.html.

TABLE 4 Cross-validation loss estimates (with standard errors inbrackets) for the Sniffer Data Set using various response distributionswith different tree depths for the mean (l) and dispersion (k). lossfunction (l, k) = (1, 0) (l, k) = (1, 1) (l, K) = (2, 1) Identity linkNormal 2.838 (0.184) 3.231 (0.18) 3.227 (0.093) Gamma 2.660 (0.119)3.056 (0.254) 2.815 (0.132) Inverse Gaussian 2.558 (0.104) 2.968 (0.084)2.735 (0.123) Canonical link Normal 2.838 (0.184) 3.231 (0.18) 3.231(0.18) Gamma 2.770 (0.137) 3.209 (0.099) 3.169 (0.050) Inverse Gaussian2.642 (0.110) 2.949 (0.221) 2.798 (0.096)

The table 4 illustrates cross-validation estimates of the negativelog-likelihood for heteroscedastic modeling (i.e., modeling to havedifferent variances) using the present invention. (The cross-validationis a way to predict a fit of a model to a hypothetical validation setwhen an explicit validation set is not available. The cross-validationestimate is an out-of-sample estimate.) A use of different tree sizes 1and k for the mean and dispersion sub-models allows different levels ofinteraction effects to be incorporated into the corresponding regressionfunctions. These results suggest that a response variance increases withthe mean. The table 4 demonstrates that the Normal model in whichpossible variable dispersion can be incorporated (i.e., for k=1) is notsuitable, nor is a case when the mean sub-model incorporates possiblefirst-order interaction terms (i.e., for k=2). In fact, the best modelfit is obtained with the Inverse Gaussian response in a constantdispersion case (independent of whether an identity link or a canonicalsquared-reciprocal link for this distribution was used in a meanregression model). The training set negative log-likelihood for themodel with l=1, k=0 was 2.32.

Synthetic Insurance Data Set

These synthetic insurance data sets are based on a Canadian AutoInsurance data set of “Two studies in automobile insurance ratemaking”,R. A. Bailey, et al, ASTIN Bulletin, Vol. 1, pp. 192-217, 1960, whollyincorporated by reference as if set forth herein, (seehttp://www.statsci.org/data/general/carinsca.html), hereinafter“Bailey”, which includes aggregated claim frequency and claims cost dataas a function of 2 covariate rating factors, viz., Merit (with 4 levels)and Class (with 5 levels), for a total of 20 cells. The web site alsodescribes GLM fits for both claims frequency (using Poisson regression)and claims cost (using Gamma regression) for models incorporating maineffects (i.e., individual terms for each covariate in an assumed form ofthe regression function). Synthetic data sets for heteroscedasticmodeling are generated using a mean dependence relationship on thecovariates from those GLM fits, along with some assumed variabledispersion dependence relationship in terms of covariate rating factors(i.e., individual levels in a Merit and Class covariate features in theInsurance data set).

A procedure for generating individual claim records in the syntheticdata sets is as follows. For each claim record, a rating factorcombination, which is a combination of a Merit and a Class ratingfactor, is selected based on an observed cell frequency (i.e., thenumber of observations of specified conditional constraints on one ormore variables) of insured claims in Bailey. The GLM model estimates forthe rating factor combination are then used to generate a Poisson randomvariate (i.e., random variable in Poisson distribution) for a claimfrequency, and Gamma random variates (i.e., random variable in Gammadistribution) for a claim cost for each of these individual claims aregenerated using an assumed Gamma dispersion parameter estimate (i.e.,estimating a dispersion parameter (φ) in Gamma distribution) for thecell. A collection of non-zero claim frequency records in this syntheticdata set is then used for heteroscedastic Gamma regression modeling(i.e., predicting one variable from one or more random variables inGamma distribution, where the one or more random variables havedifferent variance(s)), with a cost per claim being the responsevariable (i.e., an output of the heteroscedastic Gamma regressionmodeling), and the number of claims being a case weight, whichmultiplies a corresponding deviance term in the loss function.

An assumed dispersion model for a Gamma random variate generation (i.e.,generating a random variable suitable for Gamma distribution) is givenbyφ(x)=c ₁ I ₁(Class)+c ₂ I ₂(Class),  (69)so that a dispersion is a constant c₁, except in the cells with a ratingfactor Class=4, c₂ is another constant.

TABLE 5 Validation set loss estimates (with standard errors in brackets)for modeling claims costs using heteroscedastic Gamma regression for asynthetic insurance data set with a level of dispersion heterogeneitybeing varied for a same mean response (N_(Tr) denotes the number oftraining data records). c₁ = 0.5, c₂ = 1.0 c₁ = 0.5, c₂ = 2.0 N_(Tr) (l,k) = (1, 0) (l, k) = (1, 1) (l, k) = (1, 0) (l, k) = (1, 1) 500 −0.279(0.027) −0.293 (0.024) −0.222 (0.038) −0.234 (0.032) 1000 −0.308 (0.030)−0.325 (0.027) −0.222 (0.042) −0.293 (0.031) 2000 −0.305 (0.029) −0.328(0.026) −0.229 (0.040) −0.290 (0.031)

Results shown in Table (5) are all obtained using the log-link for boththe mean and dispersion sub-models. Two different pairs of values for(c₁, c₂) are considered and in each case 3 training data sets containingroughly 50, 100 and 200 records respectively were used, along with asame validation data set containing roughly 1,000 records.

The table 5 shows that it is possible to discern variations in thedispersion parameter by comparing constant dispersion models (k=0) withvariable dispersion models (k=1). For smaller contrasts in thedispersion heterogeneity (e.g., in the case c₁=0.5, c₂=1.0), largertraining data sets are required to elicit the variable dispersioneffects in a regression model.

According to one embodiment of the present invention, an extension ofthe MART methodology to joint modeling of a mean and dispersion takesinto account a fact that the mean and dispersion sub-models cannot beestimated independently, but are tightly coupled through the likelihoodformulation.

The conditional response distributions considered in one embodiment ofthe present invention, viz., the Normal, Gamma and Inverse Gaussian, aremembers of an exponential dispersion family for which the negativelog-likelihood has a special form, e.g., the formula (12). However, anoverall methodology can also be used for other conditional responsedistributions in the small dispersion limit φ→0, for which aleading-order term of the saddle-point approximation to the negativelog-likelihood also has the form (11).

In real data sets (as opposed to simulated data sets), it is often acase that a choice of basis functions for the mean regression cannotcapture a variation in a true mean, and a resulting systematicdepartures in the regression model for the mean can lead to a spuriousinference of variability in the dispersion sub-model. A carefulcross-validation analysis, along with an examination of model fits usingdifferent tree sizes in stage basis functions, may be required in orderto rule out such spurious inferences.

For a mean regression modeling, the MART methodology has certainadvantages over the GLM and other comparable regression methods, andthese advantages carry over to the joint regression modeling case thatone embodiment of the present invention implements. For example, theadaptive, non-parametric (i.e., being non-parametric means that thenumber and nature of parameters are flexible and are not fixed)regression functions in the present invention makes it possible toeasily capture additive nonlinear effects as well as low-order covariateinteraction effects in the regression modeling. The present invention isunaffected by the presence of collinear covariates, or by noisecovariates, both of which are commonly encountered in high-dimensionaldata sets in applications such as insurance. Furthermore, in contrast tothe GLM and other comparable methods, the present invention easilyincorporates high-cardinality categorical covariates in the regressionmodeling without requiring any preprocessing or grouping of featurelevels in order to reduce feature cardinality for tractablecomputational modeling.

In one embodiment, the method steps in FIGS. 1-2 are implemented inhardware or reconfigurable hardware, e.g., FPGA (Field Programmable GateArray) or CPLD (Complex Programmable Logic Device), using a hardwaredescription language (Verilog, VHDL, Handel-C, or System C). In anotherembodiment, the method steps in FIGS. 1-2 are implemented in asemiconductor chip, e.g., ASIC (Application-Specific IntegratedCircuit), using a semi-custom design methodology, i.e., designing a chipusing standard cells and a hardware description language. Thus, thehardware, reconfigurable hardware or the semiconductor chip operates themethod steps described in FIGS. 1-2.

FIG. 3 illustrates an exemplary hardware configuration of a computingsystem 1600 running and/or implementing the method steps in FIGS. 1-2.The discovery manager 220, the NLDST 230 and/or the MCDT 240 may also beimplemented on the hardware configuration illustrated in FIG. 3. Thehardware configuration preferably has at least one processor or centralprocessing unit (CPU) 1611. The CPUs 1611 are interconnected via asystem bus 1612 to a random access memory (RAM) 1614, read-only memory(ROM) 1616, input/output (I/O) adapter 1618 (for connecting peripheraldevices such as disk units 1621 and tape drives 1640 to the bus 1612),user interface adapter 1622 (for connecting a keyboard 1624, mouse 1626,speaker 1628, microphone 1632, and/or other user interface device to thebus 1612), a communication adapter 1634 for connecting the system 1600to a data processing network, the Internet, an Intranet, a local areanetwork (LAN), etc., and a display adapter 1636 for connecting the bus1612 to a display device 1638 and/or printer 1639 (e.g., a digitalprinter of the like).

Although the embodiments of the present invention have been described indetail, it should be understood that various changes and substitutionscan be made therein without departing from spirit and scope of theinventions as defined by the appended claims. Variations described forthe present invention can be realized in any combination desirable foreach particular application. Thus particular limitations, and/orembodiment enhancements described herein, which may have particularadvantages to a particular application need not be used for allapplications. Also, not all limitations need be implemented in methods,systems and/or apparatus including one or more concepts of the presentinvention.

The present invention can be realized in hardware, software, or acombination of hardware and software. A typical combination of hardwareand software could be a general purpose computer system with a computerprogram that, when being loaded and run, controls the computer systemsuch that it carries out the methods described herein. The presentinvention can also be embedded in a computer program product, whichcomprises all the features enabling the implementation of the methodsdescribed herein, and which—when loaded in a computer system—is able tocarry out these methods.

Computer program means or computer program in the present contextinclude any expression, in any language, code or notation, of a set ofinstructions intended to cause a system having an information processingcapability to perform a particular function either directly or afterconversion to another language, code or notation, and/or reproduction ina different material form.

Thus the invention includes an article of manufacture which comprises acomputer usable medium having computer readable program code meansembodied therein for causing a function described above. The computerreadable program code means in the article of manufacture comprisescomputer readable program code means for causing a computer to effectthe steps of a method of this invention. Similarly, the presentinvention may be implemented as a computer program product comprising acomputer usable medium having computer readable program code meansembodied therein for causing a function described above. The computerreadable program code means in the computer program product comprisingcomputer readable program code means for causing a computer to effectone or more functions of this invention. Furthermore, the presentinvention may be implemented as a program storage device readable bymachine, tangibly embodying a program of instructions runnable by themachine to perform method steps for causing one or more functions ofthis invention.

The present invention may be implemented as a computer readable medium(e.g., a compact disc, a magnetic disk, a hard disk, an optical disk,solid state drive, digital versatile disc) embodying program computerinstructions (e.g., C, C++, Java, Assembly languages, Net, Binary code)run by a processor (e.g., Intel® Core™, IBM® PowerPC®) for causing acomputer to perform method steps of this invention. The presentinvention may include a method of deploying a computer program productincluding a program of instructions in a computer readable medium forone or more functions of this invention, wherein, when the program ofinstructions is run by a processor, the compute program product performsthe one or more of functions of this invention.

It is noted that the foregoing has outlined some of the more pertinentobjects and embodiments of the present invention. This invention may beused for many applications. Thus, although the description is made forparticular arrangements and methods, the intent and concept of theinvention is suitable and applicable to other arrangements andapplications. It will be clear to those skilled in the art thatmodifications to the disclosed embodiments can be effected withoutdeparting from the spirit and scope of the invention. The describedembodiments ought to be construed to be merely illustrative of some ofthe more prominent features and applications of the invention. Otherbeneficial results can be realized by applying the disclosed inventionin a different manner or modifying the invention in ways known to thosefamiliar with the art.

1. A computer-implemented method for performing joint modeling of a meanand dispersion of ungrouped insurance data of high cardinality in aninsurance industry, the method comprising: deriving a loss functioncharacterized over the data taking into account distributional detailsof the ungrouped insurance data; representing regression functions forthe mean and the dispersion as stagewise expansion forms, the stagewiseexpansion forms including undetermined scalar coefficients andundetermined basis functions; determining the basis functions thatmaximally correlate with a corresponding steepest-descent gradientdirection of the derived loss function; obtaining the scalarcoefficients based on a single step of multi-dimensional Newtoniteration involving a matrix calculation; modifying parameters used forthe obtaining the scalar coefficients; and making the single step of themulti-dimensional Newton iteration be the corresponding steep-descentgradient direction for the obtained scalar coefficients; completing theregression functions based on the determined basis functions andobtained scalar coefficients; concurrently estimating the mean and thedispersion by using the completed regression functions, wherein thederiving, the representing, the determining, the obtaining, themodifying, the making, the completing, and concurrently estimating arecarried out without grouping the ungrouped insurance data, wherein aprocessor coupled to a memory device performs the deriving, therepresenting, the determining, the obtaining, the modifying, the making,the completing, and concurrently estimating.
 2. The computer-implementedmethod according to claim 1 further comprises: decreasing the lossfunction with the obtained scalar coefficients.
 3. Thecomputer-implemented method according to claim 2 further comprises:continuously halving values of the parameters until the obtained scalarcoefficients reach specific values that decreases the loss function. 4.The computer-implemented method according to claim 1, wherein thedetermined basis functions are regression trees computed using aleast-square splitting criterion.
 5. The computer-implemented methodaccording to claim 4, wherein when a depth of the regression trees is 1,the regression functions are piecewise-constant, additive and non-linearfunctions.
 6. The computer-implemented method according to claim 4,wherein when a depth of the regression trees is larger than 1, theregression functions are piecewise-constant, additive and non-linearfunctions having low-order interaction effects.
 7. Thecomputer-implemented method according to claim 4, wherein the regressiontree represents that collinear covariates are excluded from thecompleted regression functions.
 8. A non-transitory computer readablemedium embodying computer program instructions being run by a processorfor causing a computer to operate method steps for performing jointmodeling of a mean and dispersion of insurance data in an insuranceindustry, said method steps comprising the steps of claim
 1. 9. Acomputer-implemented system for performing joint modeling of a mean anddispersion of insurance data in an insurance industry, the systemcomprising: a memory device; and a processor unit in communication withthe memory device, the processor unit performs steps of: deriving a lossfunction characterized over the data taking into account distributionaldetails of the ungrouped insurance data; representing regressionfunctions for the mean and the dispersion as stagewise expansion forms,the stagewise expansion forms including undetermined scalar coefficientsand undetermined basis functions; determining the basis functions thatmaximally correlate with a corresponding steepest-descent gradientdirection of the derived loss function; obtaining the scalarcoefficients based on a single step of multi-dimensional Newtoniteration involving a matrix calculation; modifying parameters used forthe obtaining the scalar coefficients; and making the single step of themulti-dimensional Newton iteration be the corresponding steep-descentgradient direction for the obtained scalar coefficients; completing theregression functions based on the determined basis functions andobtained scalar coefficients; concurrently estimating the mean and thedispersion by using the completed regression functions, wherein thederiving, the representing, the determining, the obtaining, themodifying, the making, the completing, and concurrently estimating arecarried out without grouping the ungrouped insurance data.
 10. Thecomputer-implemented system according to claim 9, wherein the processorunit further performs a step of: decreasing the loss function with theobtained scalar coefficients.
 11. The computer-implemented systemaccording to claim 10, wherein the processor unit further performs astep of: continuously halving values of the parameters until theobtained scalar coefficients reach specific values which decreases theloss function.
 12. The computer-implemented system according to claim 9,wherein the determined basis functions are regression trees computedusing a least-square splitting criterion.
 13. The computer-implementedsystem according to claim 12, wherein when a depth of the regressiontrees is 1, the regression functions are piecewise-constant, additiveand non-linear functions.
 14. The computer-implemented system accordingto claim 12, wherein when a depth of the regression trees is larger than1, the regression functions are piecewise-constant, additive andnon-linear functions having low-order interaction effects.
 15. Thecomputer-implemented system according to claim 12, wherein theregression tree represents that collinear covariates are excluded fromthe completed regression functions.